# Nonconstant

A function is called nonconstant if it takes more than one value (if there is more than one element in its range). For example, the polynomial $p(x) = x^2 - x + 1$ with the real numbers as domain and codomain is nonconstant. We can show this simply by noting that $p(1) = 1$ and $p(2) = 3$, so the function takes at least two different values. However, the function $f: \mathbb{Z} \to \mathbb{Z}$ such that $f(x) = 1$ for all $x$ is a constant function, as the value of the function remains the same regardless of its argument, i.e. there is only one element in the codomain.

Note that recognizing nonconstant functions is not always trivial. For example, the function $f: \mathbb{Z} \to \mathbb{Z}$ that takes an integer $x$, computes the value of $x^5 -2x^4 -2x^3 - x^2 + x + 4$ and then takes the remainder of this number on division by 3 appears quite complicated but turns out to be identical to the last function in the previous paragraph: it only takes the value 1.